# Questions tagged [eigensystem]

An eigenvector of an operator is a vector such that the action of the operator is the same as multiplication by a constant, called the eigenvalue. The eigensystem of an operator is the set of all such eigenvectors and their associated eigenvalues.

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### finite difference frequency domain Eigenvalue matrix to get eigenmodes

I am using the FDFD method to calculate eigenmodes for an empty wave guide. It's a 1x1meter structure with a PEC boundary. (Here I have 6x6 points to make it simple). Sounds simple, but I can't get it ...
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### Quality of eigenvalue approximation in Lanczos method

I try to familiarize myself with iterative eigenvalue solvers such as Lanczos. So I tried rewrite it to python directly according to wiki. But it doesn't seem to work. The problem: it approximates ...
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### Fastest way to diagonalize square matrices with c++

I am currently writing software in c++ which solves the eigenvalue problem of sparse hermitian matrices. The size of the matrix depends on the user input, but as an estimation it will roughly be ...
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### Smallest eigenvalue without inverse

Suppose $A\in\mathbb{R}^{n\times n}$ is a symmetric, positive definite matrix. $A$ is big enough that it's expensive to solve $Ax=b$ directly. Is there an iterative algorithm for finding the ...
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### Numerical computation of Perron-Frobenius eigenvector

I would like to efficiently and (to the extent possible) reliably find the Perron-Frobenius eigenvector of non-negative matrices. These are not stochastic matrices, they are typically dense, and their ...
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### Memory requirement to find eigenvalues and -vectors of large sparse matrix

How can I estimate how much memory will be needed to find eigenvalues and eigenvectors of a given large sparse matrix? I have a real symmetric matrix with roughly $5 \times 10^4$ rows and columns, ...
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### Eigenvalue-style optimization with quadratic constraints

Suppose $A\in\mathbb{R}^{n\times n}$ is symmetric and positive definite and that we have several symmetric matrices $B_i\in\mathbb{R}^{n\times n}$ that are low-rank and indefinite. I need an ...
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### Does this partial eigen-expansion have a name?

This question is a follow-up to this one. Let $A\in \mathbb{R}^{n\times n}$ be large, sparse, symmetric and positive definite. Suppose for I already know $m<n$ eigenpairs of $A$, corresponding to ...
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### Julia: ordschur command

I just started leaning Julia, and my command of it is still very preliminary. I am trying to reorder a Schur decomposition. In Matlab, I could use the ordqz command and just had to specify the ...
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### last column of SPD matrix given it's spectral decomposition

I'm working on this application where I get the spectral decomposition (O,D) of a matrix A for free (A is of full rank) but not A itself, and I need to only recover the last column of A. I was ...
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### Dominant contributions of a quadratic form

Let $\Sigma$ be a covariance matrix (e.g. symmetric positive definite). For arbitrary vectors $\epsilon$, I need to compute $\chi^2 \equiv \epsilon^\top\Sigma^{-1}\epsilon$, which I do using a ...
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### Locally conservative method for differential generalized eigenvalue problem

I have to approximate the smallest eigenvalue of the following generalized eigenvalue problem $$- \nabla \cdot D(x) \nabla p(x) + \alpha(x) p(x) = \lambda \beta(x) p(x)$$ over a domain like ...
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### Is there guaranteed global solver for such an eigenvalue problem?

The original nonlinear optimization problem I have is as follows: For constant symmetric matrices $A=A^T, B_i=B_i^T(\forall i\in\mathbb{N}) \in \mathbb{R}^{n\times n}, \text{rank}(A)=n,$ \arg\min\...